Home›Calculus 1›Derivatives of logarithms›Problem 2.10

Derivative of \( \displaystyle x \ln{\left(x \right)} - x \)

Problem 2.10 · hard

Differentiate \( \displaystyle f(x) = x \ln{\left(x \right)} - x \).

\[ \frac{d}{d x} \left(x \ln{\left(x \right)} - x\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(x \right)} \]
sumApply the sum rule for differentiation.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(x \right)} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} \frac{d}{d x} x - 1 \]
productApply the product rule to the first term.✓ Proved
\[ = \ln{\left(x \right)} \]
logarithmic algebra simplifyThe derivative of log(x) is 1/x. Simplify the product x*(1/x). Final simplification.✓ Proved

Answer \( \log{\left(x \right)} \)

Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	line 1 is the problem as stated
2	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

deepseek-r1:70b: pass
gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (14)

qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies two rules at once—replacing both Derivative(x,x) with 1 and Derivative(log(x),x) with 1/x—yet only labels the logarithmic rule. Each step must change only one rule, and the label must match the rule applied.
qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 3 is labeled 'constant', but it applies the power rule (or derivative of x) to compute d/dx(x) = 1. The label 'constant' typically refers to the rule that the derivative of a constant is zero, or pulling out constant factors, neither of which describes differentiating the variable x.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (style) 2026-09-19 — Step 3 incorrectly labels the derivative of x as "constant"; the correct rule name is "derivative".
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: fail (error) 2026-09-19 — Step 3 incorrectly labels the derivative of x as 'constant' instead of 'derivative'.
gpt-oss:20b: fail (style) 2026-09-19 — Step 3 labels the derivative of x as "constant"; the correct rule name is "derivative".
qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 3 is labeled 'constant', but it applies the power rule to differentiate x into 1. The label 'constant' is incorrect for this operation.
deepseek-r1:70b: fail (misleading) 2026-09-18 — Step 3 incorrectly labels the derivative of x as 'constant'; it should be 'derivative'.
gpt-oss:20b: pass 2026-09-18
deepseek-r1:70b: pass 2026-09-16

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by gemma4:26b, checked 2026-09-26 with SymPy 1.14.0.